Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(419,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.419");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(3.68908857338\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.342102016.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 3\nu^{5} + 5\nu^{4} + 2\nu^{3} - 2\nu^{2} + 8 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} + 8\nu^{3} - 10\nu^{2} + 24\nu - 8 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + \nu^{5} - 3\nu^{4} - 6\nu^{3} - 10\nu^{2} + 8\nu - 8 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - \nu^{5} - 3\nu^{4} + 6\nu^{3} - 10\nu^{2} - 8\nu - 8 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} + 6\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} - 3\nu^{5} + 5\nu^{4} - 2\nu^{3} - 2\nu^{2} + 8 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{2} - 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{3} + 4\beta_{2} + 5\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -6\beta_{7} - 7\beta_{6} - 3\beta_{5} - 3\beta_{4} - 6\beta_{2} - 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4\beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{3} + 4\beta_{2} - 21\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(211\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 419.1 |
|
− | 1.00000i | −1.51022 | + | 0.848071i | −1.00000 | 1.69614 | 0.848071 | + | 1.51022i | −2.56155 | + | 0.662153i | 1.00000i | 1.56155 | − | 2.56155i | − | 1.69614i | ||||||||||||||||||||||||||||||||
| 419.2 | − | 1.00000i | −0.468213 | − | 1.66757i | −1.00000 | −3.33513 | −1.66757 | + | 0.468213i | 1.56155 | + | 2.13578i | 1.00000i | −2.56155 | + | 1.56155i | 3.33513i | ||||||||||||||||||||||||||||||||||
| 419.3 | − | 1.00000i | 0.468213 | + | 1.66757i | −1.00000 | 3.33513 | 1.66757 | − | 0.468213i | 1.56155 | − | 2.13578i | 1.00000i | −2.56155 | + | 1.56155i | − | 3.33513i | |||||||||||||||||||||||||||||||||
| 419.4 | − | 1.00000i | 1.51022 | − | 0.848071i | −1.00000 | −1.69614 | −0.848071 | − | 1.51022i | −2.56155 | − | 0.662153i | 1.00000i | 1.56155 | − | 2.56155i | 1.69614i | ||||||||||||||||||||||||||||||||||
| 419.5 | 1.00000i | −1.51022 | − | 0.848071i | −1.00000 | 1.69614 | 0.848071 | − | 1.51022i | −2.56155 | − | 0.662153i | − | 1.00000i | 1.56155 | + | 2.56155i | 1.69614i | ||||||||||||||||||||||||||||||||||
| 419.6 | 1.00000i | −0.468213 | + | 1.66757i | −1.00000 | −3.33513 | −1.66757 | − | 0.468213i | 1.56155 | − | 2.13578i | − | 1.00000i | −2.56155 | − | 1.56155i | − | 3.33513i | |||||||||||||||||||||||||||||||||
| 419.7 | 1.00000i | 0.468213 | − | 1.66757i | −1.00000 | 3.33513 | 1.66757 | + | 0.468213i | 1.56155 | + | 2.13578i | − | 1.00000i | −2.56155 | − | 1.56155i | 3.33513i | ||||||||||||||||||||||||||||||||||
| 419.8 | 1.00000i | 1.51022 | + | 0.848071i | −1.00000 | −1.69614 | −0.848071 | + | 1.51022i | −2.56155 | + | 0.662153i | − | 1.00000i | 1.56155 | + | 2.56155i | − | 1.69614i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 462.2.g.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 462.2.g.e | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 462.2.g.e | ✓ | 8 |
| 21.c | even | 2 | 1 | inner | 462.2.g.e | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.2.g.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 462.2.g.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 462.2.g.e | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 462.2.g.e | ✓ | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 14T_{5}^{2} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} + 2 T^{6} + \cdots + 81 \)
|
| $5$ |
\( (T^{4} - 14 T^{2} + 32)^{2} \)
|
| $7$ |
\( (T^{4} + 2 T^{3} - 2 T^{2} + \cdots + 49)^{2} \)
|
| $11$ |
\( (T^{2} + 1)^{4} \)
|
| $13$ |
\( (T^{4} + 46 T^{2} + 512)^{2} \)
|
| $17$ |
\( (T^{4} - 28 T^{2} + 128)^{2} \)
|
| $19$ |
\( (T^{4} + 74 T^{2} + 1352)^{2} \)
|
| $23$ |
\( (T^{2} + 36)^{4} \)
|
| $29$ |
\( (T^{2} + 4)^{4} \)
|
| $31$ |
\( (T^{4} + 20 T^{2} + 32)^{2} \)
|
| $37$ |
\( (T^{2} - 68)^{4} \)
|
| $41$ |
\( (T^{4} - 92 T^{2} + 2048)^{2} \)
|
| $43$ |
\( (T^{2} - 14 T + 32)^{4} \)
|
| $47$ |
\( (T^{4} - 92 T^{2} + 2048)^{2} \)
|
| $53$ |
\( (T^{2} + 4)^{4} \)
|
| $59$ |
\( (T^{4} - 266 T^{2} + 17672)^{2} \)
|
| $61$ |
\( (T^{4} + 14 T^{2} + 32)^{2} \)
|
| $67$ |
\( (T + 12)^{8} \)
|
| $71$ |
\( (T^{4} + 324 T^{2} + 20736)^{2} \)
|
| $73$ |
\( (T^{4} + 184 T^{2} + 8192)^{2} \)
|
| $79$ |
\( (T^{2} + 22 T + 104)^{4} \)
|
| $83$ |
\( (T^{4} - 74 T^{2} + 1352)^{2} \)
|
| $89$ |
\( (T^{4} - 224 T^{2} + 8192)^{2} \)
|
| $97$ |
\( (T^{4} + 40 T^{2} + 128)^{2} \)
|
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